174 XIII INTEGRATION The function /# is measurable (13.9.8.1) and we have ) £/(*)$<*) ^ almost everywhere. Hence /# is integrable (13.9.13), and the inequalities (13.12.2.1) follow. On the other hand, the function Mm(f)g —fg is defined almost everywhere and equal to (Mn(f)—f)g, hence is almost every- where ^0 in X. Hence the relation J (M^/) —f)gdn = 0 implies that (MooC/) —f)g is negligible, which completes the proof. In particular, if A is any integrable set, we have (13.12.2.2) m^/MA) g f /dp £ A For any function /on X with values in R or C, we put N^/) = Mw(\f\) if H*Q9 and NJ/) = 0 if ju = 0. By virtue of (13.12.1), the set.J??(X,/i) (resp. ^^(X, /z)) of real-valued (resp. complex- valued) measurable functions on X such that N^/) < -f oo is a real (resp. complex) vector space, and Nro is a seminorm on this space. The set of functions / such that N «.,(/) = 0 is once again the vector subspace Jf of negligible functions. The quotient of 3? £(X, u) (resp. J^c^X, /*)) by this subspace is therefore the space of equivalence classes f of measurable functions bounded in measure. It is de- noted by L£(X, ju), or simply by L£Gu) or L£ (resp. L£(X, /x), or Lg*0*) or L£). The number N^C/) is the same for all functions /belonging to the same class /e LR (resp. /e LC); it is denoted also by N «,(/), and the function f) is a worm on L£ (resp. L£). Clearly we have L% = L£ © /Lg3 . (13.12.3) For a sequence (/„) in £?$ (resp. jSf^) ro converge to a function f, it is necessary and sufficient that fn(x) should tend uniformly to f(x) in the complement of a negligible set. The condition is clearly sufficient. Conversely, if lim N^/ — /„) — 0, then n-* oo for each integer m there exists a negligible set Hm and an integer n0 such that, for all 7i ^ «OJ we have |/(jc) -/B(JC)| g l/m for all ^ e ()Hm. The union H of the sets Hm is negligible, and fn(x) tends to /(x) uniformly in (}H. (13.12.4) J/ze normed space L£(X, /i) (resp. L£(X, M)) w complete (i.e., a Banach space). Let (/„) be a sequence such that (/w) is a Cauchy sequence in L^P (resp. LC). For each integer n*tl, there exists an integer kn such thatave wsjs g w,/t for 0 <l ^ ^ t. If 0 < p < 1, use